Beta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a

For fixed b

For rational variables ||| For rational variables

Values at fixed points

General characteristics

Domain and analyticity

Β(a,b) is an analytical function of a and b which is defined in (ℂ^2) with the exception of countably many points a=-k/;k∈ℕ and b=-l/;l∈ℕ.

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to a

For fixed b  (b≠-a-m/;m∈ℕ), the function Β(a,b) has an infinite set of singular points:
a) a==-k/;k∈ℕ, are the simple poles with residues (((((-1))^k)Γ(b))/(k!Γ(b-k)));
b) a==(OverTilde[∞]) is the point of convergence of poles,  which is similar to considering (OverTilde[∞]) as an essential singular point.

With respect to b

For fixed a  (a≠-b-m/;m∈ℕ), the function Β(a,b) has an infinite set of singular points:
a) b==-k/;k∈ℕ, are the simple poles with residues (((((-1))^k)Γ(a))/(k!Γ(a-k)));
b) b==(OverTilde[∞]) is the point of convergence of poles,  which is similar to considering (OverTilde[∞]) as an essential singular point.

Branch points

With respect to a

The function Β(a,b) does not have branch points with respect to a.

With respect to b

The function Β(a,b) does not have branch points with respect to b.

Branch cuts

With respect to a

The function Β(a,b) does not have branch cuts with respect to a.

With respect to b

The function Β(a,b) does not have branch cuts with respect to b.

Series representations

Generalized power series

Expansions at generic point a==(Subscript[a, 0])

Expansions at a==0

Expansions at generic point b==(Subscript[b, 0])

Expansions at b==0

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Product representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Differentiation

Low-order differentiation

With respect to a

With respect to b

Symbolic differentiation

With respect to a

With respect to b

Fractional integro-differentiation

With respect to a

With respect to b

Integration

Indefinite integration

Involving only one direct function with respect to a

Involving one direct function and elementary functions with respect to a

Involving only one direct function with respect to b

Involving one direct function and elementary functions with respect to b

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Zeros

Theorems

Transcendentality of the beta function

For all positive rational noninteger α and β,  the value of Β(α, β) is transcendental.

The determinant of one matrix with beta functions

The determinant of the matrix A=(Subscript[{(Subscript[a, i j])}, 1≤i,j≤n]) with entries (Subscript[a, i j])=1/Β(i,j) is Γ(n).

History

– J. Wallis (1655); L. Euler (1730); J. Stirling (1730)
– A. M. Legendre  (1826) used the name "Euler function"
– P. M. Binet (1839) used the name "Beta function"
– J. J. Schönholzer (1877) derived a contour integral representation
– U. Bigler (1888) found integral representations for complex arguments

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